
theorem FAut:
for p being Prime
for n being non zero Nat
for F be Field st card F = p|^n
holds the set of all f where f is Automorphism of F =
      { (Frob F)`^m where m is Nat : 0 <= m & m <= n-1 }
proof
let p be Prime, n being non zero Nat, F be Field;
assume AS1: card F = p|^n; then
AS2: F is finite;
set P = PrimeField F;
reconsider E = F as P-finite FieldExtension of P by AS2,FA1;
consider a being Element of E such that
A: E == FAdj(P,{a}) by AS2,FIELD_14:def 7;
set M = the set of all f where f is P-fixing Automorphism of FAdj(P,{a});
reconsider M as finite set by FAUTfin;
C: card {(Frob F)`^m where m is Nat : 0 <= m & m <= n-1} = n by AS1,Frobcard;
then reconsider K = {(Frob F)`^m where m is Nat : 0 <= m & m <= n-1}
     as finite set;
set m = card M;
   C1: deg(FAdj(P,{a}),P) = deg(E,P) by A,FIELD_7:5 .= n by AS1,FA3;
   C2: m <= card(Roots(FAdj(P,{a}),MinPoly(a,P))) by FAUTh;
   C3: card(Roots(FAdj(P,{a}),MinPoly(a,P))) <= deg MinPoly(a,P) by ID2;
       deg MinPoly(a,P) = n by C1,FIELD_6:67; then
D: m <= n by C2,C3,XXREAL_0:2;
E: M = the set of all f where f is Automorphism of FAdj(P,{a})
   proof
   E1: now let o be object;
       assume o in the set of all f where f is Automorphism of FAdj(P,{a});
       then consider f being Automorphism of FAdj(P,{a}) such that
       E2: o = f;
       PrimeField FAdj(P,{a}) = PrimeField F by RING_3:94; then
       f is P-fixing  by AS2,FA4;
       hence o in M by E2;
       end;
   now let o be object;
       assume o in M;
       then consider f being P-fixing Automorphism of FAdj(P,{a}) such that
       E3: o = f;
       thus o in the set of all f where f is Automorphism of FAdj(P,{a}) by E3;
       end;
   hence thesis by E1,TARSKI:2;
   end;
B: K c= M
   proof
   now let o be object;
    assume o in K; then
    consider m being Nat such that B1: o = (Frob F)`^m & 0 <= m & m <= n-1;
    reconsider f = (Frob F)`^m as Automorphism of FAdj(P,{a}) by AS2,A,FAutEQ1;
    f is P-fixing by AS2,FA4;
    hence o in M by B1;
    end;
   hence thesis;
   end;
then n <= m by C,NAT_1:43;
then K = M by B,C,D,XXREAL_0:1,FIELD_10:4;
hence thesis by E,A,FAutEQ;
end;
